Encyclopedia of thermal stresses

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  • 1. 1 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics Synonyms Alumina; Distinct element method; Internal residual stresses; Polycrystalline ceramics; Sintering Overview Anisotropic alumina grains is the main reason of occurrence of internal residual stressesthermal expansion coefficient of during in polycrystalline alumina ceramics. Investigation of the effect of internal residual stresses on the crack-tipsintering toughness of alumina polycrystalline ceramics presents a matter of this entry. Crack-tip toughness has a significant role in the mechanical performance of alumina structures because it affects the stress value at which the crack starts to propagate, and it also determines the strength of alumina in the case of existence of small cracks. The relationship between the grain size, internal residual stresses, and the crack-tip toughness of polycrystalline ceramics (alumina) is the main subject of this entry. Application of the distinct element method is discussed for the numerical prediction of(DEM) effect of residual stresses on crack-tip toughness. Introduction Polycrystalline ceramic materials generally exhibit both internal (bulk) and surface residual stresses [ , ]. Internal1 2 residual stresses, which arise during sintering, occur primarily due to the inhomogeneous withintemperature distribution the sample and the anisotropic thermal caused by randomly oriented crystals.expansion coefficient Surface treatment such as mechanical and introduces residual stresses on the surface of the ceramics due to differentialgrinding polishing between the damaged and undamaged regions [ ]. Such residual stresses affect the mechanicalpermanent deformation 3 performance (e.g., strength, ) of ceramic materials. Thereof, it is important to measure the residualfracture toughness stresses and to investigate the effect of these stresses on mechanical properties of ceramics. In this entry, the effect of internal residual stresses on the crack-tip toughness of alumina ceramics is discussed. Surface residual stresses occur due to machining of the ceramic structures. During abrasive grinding, materialsabrasive together with the flaws in the surface region are removed, while residual stresses are introduced into the newly formed surface region. The presence of a surface is normally thought to be a result of local . Inresidual stress plastic deformation polycrystalline alumina, diffusion-driven grain boundary sliding is possibly an important deformation mechanism [ ]. The4 magnitude of surface residual stresses can be measured by using X-ray topography technique, photoelastic technique, indentation technique, X-ray diffraction technique, bending technique, and confocal Cr3+ fluorescence microscopy [ ].5 Thermal stresses result from thermal expansion incompatibility in materials during the temperature change. Main reasons of thermal stresses are thermomechanical mismatch between two materials, nonuniform temperature distribution in a material, and thermal expansion anisotropy in a single phase material. The first one appears due tophase transformation thermal expansion mismatch between dissimilar materials (e.g., composites, multiphase structures, and layered composite structures). The second one occurs due to nonuniform heating/cooling of surface and interior part of the material. Thermal stresses due to phase transformation occur, for example, in pure or stabilized zirconia, and itzirconia results from volume changes during cooling due to cubic-to-tetragonal-to-monoclinic phase transformation. The main motivation of this entry is about the last type of thermal stress occurrence. The last one exists in single phase polycrystalline materials at the microscopic level where localized thermal stresses arise from thermal expansion mismatch among randomly oriented grains. Thermal stresses due to thermal anisotropy arise because of different thermal expansion coefficients in the radial and tangential directions in a cylindrical geometry during temperature change. For a material with crystal symmetry lower than cubic, the thermal expansion coefficient, α, can differ along different crystal directions. For example, in Fig. 1, the two-dimensional representation of a tetragonal or hexagonal crystal structure is shown. Grains are randomly distributed, and they have random directions (i.e., random -direction). In such a case,c during cooling, stresses develop at grain boundaries. In brittle materials, some part of these stresses causes microcracks
  • 2. 2 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 within or between individual grains, and the rest stay in the material as residual stresses (tension or compression stresses). Fig. 1 Two-dimensional representation of tetragonal or hexagonal crystal structure In polycrystalline alumina, which is a non-cubic and non-transforming material, internal residual stresses occur during due to the thermal expansion anisotropy of constituent alumina crystals. These residual stresses are reported tosintering be the main reason of the dependence of crack-tip toughness ( ) on average grain size. In alumina, the average grainKI0 shape is hexagonal. The thermal in the vertical direction ( -direction) of hexagonal grain is differentexpansion coefficient c from the horizontal direction ( -direction) of the grain [ ]. This results in a different shrinkage rate in different directions ofa 6 grains. This difference causes some thermal stresses (so-called internal residual stresses) after shrinkage in alumina. In solid state sintering of ceramics, shrinkage is driven by the surface energy of the porous compacts. The grain boundary diffusion and volume diffusion are the main material transfer mechanisms which lead to shrinkage of compacts [ ]. Often7 densification in alumina is attributed to a control by grain boundary diffusion where the dominant path for diffusion is usually along the grain boundary [ ]. In this mechanism, it is assumed that the energy provided is available to drive the8 along the grain boundaries. However, Burton [ ] stated that some energy is expended for materials to bediffusional flux 9 added to (or removed from) a grain boundary when an interface reaction occurs. He and Ma [ ] investigated the effect of10 grain size on the dominant densification mechanism by using alumina powders with average grain size of 0.9 and 7.0 μm. They concluded that for small grain sizes, the interface reaction becomes the controlling process for sintering, and therefore, the transport path becomes shorter. As the grain size increases, the distance that materials have to travel along grain boundary increases, and the grain boundary diffusion becomes more important. Therefore, since the alumina with small grain size can have more time for the down to lower temperatures, when cooled from a stress-freestress relaxation processing temperature, smaller residual stresses are expected as compared to an alumina with larger grain size [ ].11 The effect of heating rates on densification rates can be determined by using the kinetic field diagrams. The kinetic field diagram is unique for a special type of green body. Therefore, when the grain size is changed, the kinetic field diagram has to be rebuilt. In general, the sintering rate (densification rate) increases with decreased particle size and with increased sintering temperature and time. Furthermore, the grain growth increases with decreasing grain size [ ]. Saha12 et al. [ ] suggested low heating rate for low anisotropy which means lower residual stresses considering the13 coefficient anisotropy. More details about the thermodynamical background of the densification of denseof thermal expansion powder compacts, coarsening of grains and pores during the densification, are given by Lange [ ].14 As it is mentioned above, there is a strong relationship between the average grain size, internal residual stresses which occur due to anisotropic thermal alumina grains in different directions and the crack-tip toughnessexpansion coefficient of of polycrystalline alumina ceramics. Recently, there has been increasing attention given to crack-tip toughness (critical stress intensity factor) which determines the initiation of crack propagation and the strength of the ceramics in case of existence of small cracks. In the following, this relationship is described, and the effect of internal residual stresses on the crack-tip toughness of alumina is discussed according to experimental measurements. Afterward, a numerical model prepared by distinct element method will be used in order to show the variation of crack-tip toughness with internal(DEM) residual stresses. Failure of Advanced Ceramics At low and ambient temperatures, failure of ceramics is always brittle and occurs without any significant plastic deformations. occurs due to unavoidable presence of flaws that results either during cooling from the meltBrittle fracture
  • 3. 3 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 (volume cracks) or machining process of ceramic materials (surface cracks). These preexisting cracks cause a local at the crack-tip which is several times higher than the applied stress, and the crack grows when thestress concentration stress at crack-tip is equal to the theoretical strength. Size, position, and orientation of these cracks show variation from specimen to specimen. This variation causes a scattering of strength of ceramic materials. Therefore, strength is not a definite value for ceramics. Instead, stress intensity factor is used for the analysis of fracture of ceramic materials [ ].15 The stress intensity factor is commonly used as a measure of the driving force for crack propagation and is used in to predict the stress state near the tip of a crack caused by a remote load or residual stresses. It is afracture mechanics theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials. The stress intensity factor, , is a function of loading, crack size, and structural geometry. The stressK intensity factor is calculated by the following equation: (1) where is the stress intensity factor, is the crack length, is the applied stress, and is the geometric functionKI a σA Y characterizing the influence of the body, crack size and shape, and type of loading. A Roman numeral subscript (I) indicates the mode-I fracture. Mode-I fracture is the condition in which the crack plane is normal to the direction of largest tensile loading. This is the most commonly encountered mode, and therefore, for the remainder of the material, we will consider . In , the stress intensity factor increases with increasing the applied load untilKI linear elastic fracture mechanics a critical value of is reached, at which the crack is still in equilibrium and above which unstable crack propagation takesKI place. This critical value is called and termed . The unstable crack propagation occurs when =fracture toughness KIC KI K . Fracture toughness is an indication of the amount of stress required to propagate a preexisting flaw. In the theory ofIC fracture discussed until now, it was assumed that the unstable crack propagation occurs when the condition = isKI KIC satisfied. In many ceramics, a different behavior can be observed that the crack growth resistance increases with increasing crack extension due to so-called toughening mechanisms (e.g., crack and orbridging phase transformation microcracking zone) [ ]. In other words, stable crack propagation occurs preceding the unstable failure.16 The crack propagation behavior is no more characterized by a single value but with which increases fromKIC KIR KI0 which is the onset value of crack growth (crack-tip toughness). In Fig. 2, an idealized R-curve behavior is represented. This figure also shows a series of stress intensity factor curves (shown with dashed lines) for several levels of applied load ( < < ) which result in a stable crack growth of Δ and Δ .P1 P2 P3 a1 a2 Fig. 2 versus Δ curve for a material with a rising crack resistanceK IR a Crack-tip toughness is an important parameter because it determines the crack propagation, and it is the strength of the ceramic materials which contains small cracks.
  • 4. 4 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 Relationship Between Average Grain Size, Internal Residual Stresses, and Crack-Tip Toughness There is a strong relationship between grain size, residual stresses, and the crack-tip toughness of alumina. An increase in residual stresses causes a decrease of crack-tip toughness. Increase of residual stresses is a result of increase of average grain size of alumina. This relationship is explained and discussed in this part regarding the experimental measurements. Seidel and Rödel [ ] measured of alumina as a function of average grain size by using crack profile measurements17 KI0 with . Njiwa et al. [ ] used crack opening displacement (COD) andscanning electron microscopy (SEM) 18 strain gauge methods and observed a decreasing trend of with increasing average grain size. It was reported that an increase inKI0 the residual stresses may increase the damage at the crack-tip stress field of a growing macrocrack in the form of a microcrack zone, deteriorate the local fracture resistance at the crack-tip, and lead to a decrease of with an increaseKI0 in grain size [ ].18 In Fig. 3, the experimentally measured results are represented. The data points are approximated by a best line. Since strain gauge method overmeasures the due to poor quality of the amplifier, COD measurements are used asKI0 reference. Fig. 3 Experimentally measured dependence of crack-tip toughness on average grain size according to Seidel and Rödel [ ] and+ 17 Njiwa et al.* [ ] (Reproduced with permission from [ ])18 23 The relationship between the grain size, flaw size, and residual stresses was introduced by an expression for the first time by Krstic [ ]. Krstic [ ] reported that at small grain sizes (below approximately several microns), the19 19 residual stress plays a minor role in determining the strength level in the material. As the grain size increases, above approximately 10 μm, the effect of residual stress becomes more important, and at very large sizes (>100 μm), it dominates the fracture of the brittle materials. Krell and Grigoryev [ ] used X-ray diffraction data and computed a mean residual stress in -direction of < > =20 c σres,c 40–70 MPa for an average grain size of 3 μm and a level of < > =20–30 MPa for grain sizes below 1 μm. Krell et al.σres,c [ ] reported that the residual stresses take value of < > = 30–100 MPa for grain sizes of 1–9 μm and < >21 σres,c σres,c =20–30 MPa for grain sizes smaller than 1 μm. By applying the technique of piezospectroscopy using the fluorescence from trace Cr3+ impurities, the residual stresses change < > =130–270 MPa for a grain size regime of 2–20 μm [σres,c 22 ]. Figure 4 represents the residual stress measurements of Ma and Clarke [ ] as a function of average grain size. The22 error bars in the graph show the maximum and minimum values. By using the experimental values, the residual stresses as a function of average grain size can be predicted with the help of Evans–Clarke model [ ] as follows:11
  • 5. 5 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 (2) where < > is the average residual stress, is the Poisson’s ratio, Δα is the deviation of the contraction coefficient fromσ v average, is a coefficient that depends on the orientation of the adjacent grains and contains corrections to account forβ the effective , is the temperature, is the cooling rate, is the activation energy for diffusion, is theelastic modulus T Q k Boltzmann constant, is the diffusion parameter, Ω is the atomic volume, is the elastic modulus, and is the grainD0 δb E a size. For Al O , with = 10 m s , = 100 kcal/mole, Ω 10 m , = 30, = 420 GPa, = 0.3, and Δα = 7 × 102 3 D0 δb 9 3 1 Q = 29 3 n E v −7 1/K, average residual stress as a function of grain size is given in Fig. 5 for different fitting parameter .β Fig. 4 Measured residual stresses of alumina in -direction as a function of grain size [ ] (Reproduced with permission from [ ])c 22 23 Fig. 5 Fitting of Evans–Clarke function for alumina with different values (Reproduced with permission from [ ])β 23 For , the Evans–Clarke function gives the best fitting of experimental measurements for the investigated alumina.β Numerical DEM Model for the Investigation of Effect of Residual Stresses on Crack-Tip Toughness of Alumina In this part, distinct element method is used in order to simulate the effect of residual stresses (average grain size)(DEM) on the crack-tip toughness of alumina polycrystalline ceramics [ ]. It is a numerical mesh-free method which is used to23 simulate the motion and response of granular media and heterogeneous materials by modeling the dynamic behavior of
  • 6. 6 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 assemblies of particles (e.g., circular discs, spheres, and blocks) [ ]. In contrast with classical methods like finite24 element and boundary element methods that treat the medium as a continuum, the distinct element method treats the medium as a discontinuum. The emphasis of distinct element method is therefore in reproducing the mechanics of contacts and impacts between distinct blocks. PFC (particle flow code in 2 dimensions) is a discontinuum code used in analysis, testing, and research in any field2D where the interaction of many discrete objects exhibiting large strain and/or fracturing is required. PFC particle flow2D model simulates the mechanical behavior of a system comprised of a collection of circular distinct particles. Circular particles displace independently from one another, and they interact at contacts which occur over a ravishingly small area. The particles are assumed to be rigid, and behavior of the contact is characterized by a soft contact [ ]. The25 behavior of a heterogeneous can be modeled by the introduction of parallel bonds between the particlesbulk material which break when the stresses acting on the bonds exceed the . In the DEM simulations, it is assumed thatbond strength the heterogeneous materials consist of particles (grains in the material) and the bonds between the particles (grain boundaries in the material). In this way, the properties (e.g., Young’s modulus and strength) of the grains (particles) and the grains’ boundaries (bonds) can be defined separately. The single-edge-notched beam (SENB) test is simulated for the calculation of the . For every averagefracture toughness grain size of a model material, the corresponding internal value is computed according to theresidual stress Evans–Clarke model with = 0.55. The residual stresses enter into the model as a standard deviation to the normallyβ distributed normal and shear strength of the parallel bonds. As a result, the normal and shear strength of the parallel bonds between the grains in the model are distributed by a with a mean value of bond strength andGaussian distribution a standard deviation of residual stress. Since tensile residual stress decreases the mean strength and compressive residual stress increases the mean strength, the required stress in order to break a parallel bond will be at some bonds higher and at some bonds lower than the mean strength value. For each average grain size, seven different packing arrangements were produced since the particle arrangement may affect the behavior of the material. This difference between these seven packing arrangements was achieved during the model generation. The average value of seven simulations will be used as the main result, and the variability of these results will be given as error bars in the following. In Fig. 6, the pictures of the failed specimens are given for = 2 μm and for = 80 μm. The residual stress valuesGavg Gavg for = 2 μm and for = 80 μm are taken as 130 MPa and 430 MPa, respectively. The crack propagates betweenGavg Gavg the particles with dark colors. A small region of the 4-point bending test specimen is given in order to show the crack path. In the longitudinal direction, there exist more particles. In Fig. 6a, the crack propagates almost straightly as a primary crack. In Fig. 6b, it is possible to observe crack redirecting and secondary cracks which are connected to primary crack. Moreover, there are some microcracks, which are not connected to primary crack and appeared in the region under the tension stresses. Fig. 6 Representation of crack propagation in the specimens with average grain size of ( ) = 2 μm and ( ) = 80 μma Gavg b Gavg (Reproduced with permission from [ ])23 When the experimentally observed dependence of crack-tip toughness measured with COD method (see Fig. 3) is compared with the model results, it can be said that the model gives almost the same values and creates theDEM decreasing trend with increasing average grain size (see Fig. 7). Moreover, the experimentally measured values are inside the interval which can be shown using the error bars of the DEM model. This also shows the compatibility of experimental and numerical results.
  • 7. 7 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 Fig. 7 Comparison of the normalized values with experimentally measured ones (Reproduced with permission from [ ])KI0 23 Conclusion In this entry, the effect of thermal stresses, which occur during the to the anisotropicsintering process due thermal randomly oriented alumina crystals, on crack-tip toughness of alumina was discussed. Anexpansion coefficient of increase of average grain size of alumina results in an increase of internal residual stresses. If the residual stresses increase, the crack-tip toughness of alumina decreases. As a result, a preexisting crack in an alumina component starts to propagate at a lower stress value. A numerical distinct was used for the prediction of crack-tipelement model toughness as a function of grain size. References 1. Asokan T, Sudarshan TS (1994) Effect of residual stress on the surface flashover of alumina ceramics. IEEE T Dielect El In 1(1):97–105 2. Blendell JE, Coble RL (1982) Measurement of stress due to thermal expansion anisotropy in Al O . J Am2 3 Ceram Soc 65:174–178 3. Stokes RJ (1972) Effects of surface finishing on mechanical and other physical properties of ceramics. In: Schneider SJ, Rice RW (eds) Science of ceramic machining and surface finishing. National Bureau of standard special publication, Washington, DC, pp 343–352 4. Cannon RM, Rhodes WH, Heuer AH (1980) Plastic deformation of fine-grained alumina (Al O ): I,2 3 interface-controlled diffusional creep. J Am Ceram Soc 63:46–53 5. Tuan WH, Kuo JC (1999) Contribution of residual stress to the strength of abrasive ground alumina. J Eur Ceram Soc 19:1593–1597 6. Rahaman MN (2007) Sintering of ceramics. CRC Press, Boca Raton 7. Raether F, Horn PS (2009) Investigation of sintering mechanisms of alumina using kinetic field and master sintering diagrams. J Eur Ceram Soc 29(11):2225–2234 8. Kingery WD, Bowen HK (1975) Introduction to ceramics. Wiley, New York 9. Burton B (1972) Interface reaction controlled diffusional creep: a consideration of grain boundary dislocation climb sources. Mater Sci Eng 10:9–13 10. He Z, Ma J (2005) Constitutive modeling of alumina sintering: grain-size effect on dominant densification mechanism. Comp Mater Sci 32(2):196–202 11. Evans AG, Clarke DR (1980) Residual stresses and microcracking induced by thermal contraction inhomogeneity. In: Thermal Stress in Severe Environments. Edited by D. P H. Hasselman and R. A. Haller. Plenum, New York, pp 629–648 12. Kang SL (2005) Sintering: densification, grain growth and microstructure. Elsevier-Butterworth-Heinemann, Burlington 13. Saha BP, Johnson R, Ganesh I, Rao GVN, Bhattacharjee S, Mahajan YR (2001) Thermal anisotropy in
  • 8. 8 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 sintered cordierite monolith. Mater Chem Phys 67:140–145 14. Lange FF (2008) Densification of powder compacts: an unfinished story. J Eur Ceram Soc 28(7):1509–1516 15. Munz D, Fett T (2001) Ceramics: mechanical properties, failure behaviour, materials selection. Springer, Berlin/New York 16. Llorca J, Steinbrech RW (1991) Fracture of alumina: an experimental and numerical study. J Mater Sci 26:6383–6390 17. Seidel J, Rödel J (1997) Measurement of crack tip toughness in alumina as a function of grain size. J Am Ceram Soc 80(2):433–438 18. Njima ABK, Yousef SG, Fett T, Rödel J (2005) Influence of microcracking on crack-tip toughness of alumina. Eng Fract Mech 72(7):1011–1019 19. Krstic VD (2006) Effect of microstructure on fracture of brittle materials: unified approach. Theor Appl Fract Mech 45(3):212–226 20. Krell A, Grigoryev N (1990) Residual stresses and microporosity in oxide ceramics. Sprechsaal 123:1012–1015 21. Krell A, Terersiak A, Schlaefer D (1996) Grain size dependent residual microstresses in submicron AL O and2 3 ZrO . J Eur Ceram Soc 16(8):803–8112 22. Ma Q, Clarke DR (1994) Piezospectroscopic determination of residual stresses in polycrystalline alumina. J Am Ceram Soc 76:1433–1440 23. Nohut S (2011) Prediction of crack-tip toughness of alumina for given residual stresses with parallel-bonded-particle model. Comp Mater Sci 50:1509–1519 24. Cundall PA (1971) A computer model for simulating progressive, large scale movements in blocky rock systems. In: Proceedings of international symposium on rock mechanics, Nancy II, pp 8 25. Potyondy DO, Cundall CA (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364 Further Reading Anne G, Hecht-Mijic S, Richter H, Vander Biest O, Vlengels J (2006) Strength and residual stresses of functionally graded Al O /ZrO discs prepared by electrophoretic deposition. Scr Mater 54(12):2053–20562 3 2 Buresch FE (1985) Relations between the damage in and microstructure of ceramics. Mater Sci Eng 71:187–194 Nohut S (2009) Reliability of advanced ceramics: macro- and mesoscale investigations. Cuvillier, Göttingen Paulik SW, Zimmerman MH, Faber KT (1996) Residual stress in ceramics with large thermal expansion. J Mater Res 11(11):2795–2803 Swain MV, Hannink RHJ (1984) R-curve behavior in zirconia ceramics. In: Claussen N, Rühle M, Heuer AH (eds) Advance in ceramics, vol 12, Science and technology of zirconia II. American Ceramic Society, Columbus, pp 225–239 Wachtman JB (1996) Mechanical properties of ceramics. Wiley-Interscience, New York, USA
  • 9. 9 SpringerReference Serkan Nohut Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics 11 Feb 2013 15:52http://www.springerreference.com/index/chapterdbid/364221 © Springer-Verlag Berlin Heidelberg 2013 Effect of Thermal Stresses on Crack-Tip Toughness of Polycrystalline Ceramics Serkan Nohut Faculty of Engineering, Zirve University, Gaziantep, Turkey DOI: 10.1007/SpringerReference_364221 URL: http://www.springerreference.com/index/chapterdbid/364221 Part of: Encyclopedia of Thermal Stresses Editors: - PDF created on: February, 11, 2013 15:52 © Springer-Verlag Berlin Heidelberg 2013